Suppose our probability space is $(\Omega, \mathcal{F}, \mathbb{P})$. In the lecture notes of my instructor, the following definition was given: $$ \mathbb{E}[Y(w)|X(w) = x_k] = \int Y(w)\mathbb{P}(dw|X(w)=x_k), $$ where $X$ takes on distinct values $\{x_k\}$.
I don't quite understand how to interpret $\mathbb{P}(dw|X(w)=x_k)$.
My understanding is that $\mathbb{P}(dw)$ measures an infinitesimal set $dw$ which is presumably in $\sigma$-algebra $\mathcal{F}$. When we are given $X(w)=x_k$, then our $\sigma$-algebra is reduced to events that could yield $X(w)=x_k$. So $\mathbb{P}(dw|X(w)=x_k)$ only measures sets $dw$ such that $w \in dw$ and $X(w)=x_k$? Does this make any sense?
The object $Q(A,t):=\mathbb{P}(A|X=t)$, in the context of the question, its known as a probability (or stochastic) kernel, that is, a map $Q:\mathcal{F}\times \mathbb{R}\to \mathbb{R}$, for a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ and a random variable $X:\Omega \to \mathbb{R}$, such that for every $t\in \mathbb{R}$ the map $A\mapsto Q(A,t)$ is a probability measure, and for every $A\in \mathcal{F}$ the map $t\mapsto Q(A,t)$ is a measurable function.
Its an object that is part of one of the ways that we can disintegrate the measure $P(A,B):=\mathbb{P}(A\cap \{X\in B\})$ as the product $P(d\omega ,dt)=Q(d\omega ,t)P_X(dt)$, where $P_X:=\mathbb{P}\circ X^{-1}$.
If the event $\{X=t\}$ have positive measure, then $\mathbb{P}(d\omega |X=t)$ is the probability measure $\mu$ defined by
$$ \mu(A):=\mathbb{P}(A|X=t)=\frac{\mathbb{P}(A\cap \{X=t\})}{\mathbb{P}(X=t)} $$
By example, using the above notations, we have that $$ Q(A,t)=\int_{A}Q(d\omega,t ) $$ by definition of the Lebesgue integral (that is, by definition of what means the integration respect to a measure). Hope you can see everything more clear now.